Under the ultrafilter lemma and Dependent Choice, a Tychonoff space is $G_{\delta }$ in some Hausdorff compactification exactly when it is $G_{\delta }$ in every one

Given: The objects, hypotheses, and choice principles stated above.

A Tychonoff space $X$ is Čech-complete when there is a Hausdorff compactification $(K,i)$ of $X$ (def-compactification-of-a-tychonoff-space) for which $i[X]$ is a $G_{\delta }$ subset of $K$ (def-g-delta-and-f-sigma-in-a-topological-space). The definition asks for one compactification; thm-cech-completeness-is-independent-of-compactification proves the equivalent every-compactification form. (Čech-complete spaces as $G_{\delta }$ subspaces of Hausdorff compactifications).

Let $X$ be dense in Hausdorff compactifications $K$ and $L$, and let $f:K\to L$ be continuous with $f{\mathrm{\mid }}_X={\mathrm{id}}_X$. Then $f$ is surjective and $f[K\setminus X]=L\setminus X$. (A map of Hausdorff compactifications carries the larger remainder onto the smaller remainder).

A Stone–Čech compactification of $X$ is a Hausdorff compactification $(B,i)$ (def-compactification-of-a-tychonoff-space) such that for every compact Hausdorff space $K$ and continuous map $f:X\to K$ (def-continuous-map-top), there is a unique continuous $\bar{f}:B\to K$ with $\bar{f}\circ i=f$. The universal property, rather than a particular construction, is the definition. (The Stone–Čech compactification by its compact-Hausdorff extension property).

Under the hypotheses of thm-stone-cech-evaluation-closure-universal-property, two Stone–Čech compactifications $(B,i)$ and $(B^{\mathrm{\prime }},i^{\mathrm{\prime }})$ of $X$ are uniquely homeomorphic by a map $u:B\to B^{\mathrm{\prime }}$ satisfying $u\circ i=i^{\mathrm{\prime }}$. (Stone–Čech compactifications are uniquely homeomorphic over the original space).